document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1137506>Question 1137506</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do I find the formula for an exponential function that passes through the two given points?<BR>\n" );
document.write( "(0,6) and (3,750) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #755382 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=755382\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" color=\"black\" size=\"3\"><BR>\n" );
document.write( "The general template for an exponential function is <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+a%2Ab%5Ex ALIGN=MIDDLE  ALT=\"y+=+a%2Ab%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where 'a' and 'b' are fixed values (constants).\r<BR>\n" );
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document.write( "The first point given to us is (0,6) meaning that x = 0 and y = 6 pair up together. Let's plug them both into the template to see what happens<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+a%2Ab%5Ex ALIGN=MIDDLE  ALT=\"y+=+a%2Ab%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6+=+a%2Ab%5E0 ALIGN=MIDDLE  ALT=\"6+=+a%2Ab%5E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > replace x with 0; replace y with 6\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6+=+a%2A1 ALIGN=MIDDLE  ALT=\"6+=+a%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > anything to the zeroth power is equal to 1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6+=+a ALIGN=MIDDLE  ALT=\"6+=+a\"   DISABLED_style= \"max-width:90%; height:auto;\" > one times any number is itself\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+6 ALIGN=MIDDLE  ALT=\"a+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" > So we have one of the constants we need.\r<BR>\n" );
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document.write( "Now plug in the other point (x,y) = (3,750). Also we will plug in a = 6 that we found earlier. Then solve for b\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+a%2Ab%5Ex ALIGN=MIDDLE  ALT=\"y+=+a%2Ab%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+6%2Ab%5Ex ALIGN=MIDDLE  ALT=\"y+=+6%2Ab%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > replace 'a' with 6\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=750+=+6%2Ab%5E3 ALIGN=MIDDLE  ALT=\"750+=+6%2Ab%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > plug in (x,y) = (3,750)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=750%2F6+=+%286%2Ab%5E3%29%2F6 ALIGN=MIDDLE  ALT=\"750%2F6+=+%286%2Ab%5E3%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 6\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=125+=+b%5E3 ALIGN=MIDDLE  ALT=\"125+=+b%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E3+=+125 ALIGN=MIDDLE  ALT=\"b%5E3+=+125\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2Cb%5E3%29+=+root%283%2C125%29 ALIGN=MIDDLE  ALT=\"root%283%2Cb%5E3%29+=+root%283%2C125%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Apply the cube root to both sides. You can also think of it as raising both sides to the 1/3 power. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+5 ALIGN=MIDDLE  ALT=\"b+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > We found the other constant\r<BR>\n" );
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document.write( "Therefore, <BR>\n" );
document.write( "a = 6<BR>\n" );
document.write( "b = 5\r<BR>\n" );
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document.write( "Making the function be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+6%285%29%5Ex ALIGN=MIDDLE  ALT=\"f%28x%29+=+6%285%29%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+6%285%29%5Ex ALIGN=MIDDLE  ALT=\"y+=+6%285%29%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > since y = f(x).\r<BR>\n" );
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document.write( "As a check, plug in x = 0 to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+6%285%29%5Ex ALIGN=MIDDLE  ALT=\"f%28x%29+=+6%285%29%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29+=+6%285%29%5E0 ALIGN=MIDDLE  ALT=\"f%280%29+=+6%285%29%5E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29+=+6%281%29 ALIGN=MIDDLE  ALT=\"f%280%29+=+6%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29+=+6 ALIGN=MIDDLE  ALT=\"f%280%29+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So we get the proper output for the input x = 0. Repeat for the input x = 3 as well<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+6%285%29%5Ex ALIGN=MIDDLE  ALT=\"f%28x%29+=+6%285%29%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%283%29+=+6%285%29%5E3 ALIGN=MIDDLE  ALT=\"f%283%29+=+6%285%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%283%29+=+6%28125%29 ALIGN=MIDDLE  ALT=\"f%283%29+=+6%28125%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%283%29+=+750 ALIGN=MIDDLE  ALT=\"f%283%29+=+750\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "that output is correct as well. The function is confirmed. Both points are on the graph of this function curve.<BR>\n" );
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